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Lecture 7
QUANTUM PROBABILITY
Stéphane ATTAL
Abstract Bell Inequalities and the failure of hidden variable approaches
show that random phenomena of Quantum Mechanics cannot be modeled
by classical Probability Theory. The aim of Quantum Probability Theory is
to provide an extension of the classical theory of probability which allows
to describe those quantum phenomena. Quantum Probability Theory does
not add anything to the axioms of Quantum Mechanics, it just emphasizes
the probabilistic nature of them. This lecture is devoted to introducing this
quantum extension of Probability Theory, its connection with the quantum
mechanical axioms. We also go a step further by introducing the Toy Fock
spaces and by showing how they hide very rich quantum and classical probabilistic structures.
For reading this chapter the reader should be familiar with the basic notions of Quantum Mechanics, but also of Probability Theory and discrete
time stochastic processes. If necessary read Lectures 4 and 5.
Stéphane ATTAL
Institut Camille Jordan, Université Lyon 1, France
e-mail: [email protected]
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Stéphane ATTAL
7.1
Observables and Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1
States and Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2
Observables vs Random Variables . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.3
The Role of Multiplication Operators . . . . . . . . . . . . . . . . . . . . . .
7.1.4
Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Quantum Bernoulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1
Quantum Bernoulli Random Variables . . . . . . . . . . . . . . . . . . . . .
7.2.2
The Toy Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3
Probabilistic Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4
Brownian Motion and Poisson Process . . . . . . . . . . . . . . . . . . . . .
7.3
Higher Level Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1
Structure of Higher Level Chains . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2
Obtuse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3
Obtuse Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.4
Generic Character of Obtuse Random Variables . . . . . . . . . . . . .
7.3.5
Doubly-symmetric 3-tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.6
The Main Diagonalization Theorem . . . . . . . . . . . . . . . . . . . . . . . .
7.3.7
Back to Obtuse Random Variables . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.8
Probabilistic Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1 Observables and Laws
There exists plenty of different definitions of a quantum probability space in
the literature, with different levels of generality. Most commonly Quantum
Probability Theory is defined at the level of von Neumann algebras and normal states. In this lecture we consider a lower level of generality in order to
stick to the most usual axioms of Quantum Mechanics, that is, the Hilbert
space level, where states are density matrices and observables are self-adjoint
operators.
7.1.1 States and Observables
Definition 7.1. A quantum probability space is a pair (H, ρ) where H is a
separable Hilbert space and ρ is a density matrix on H. Such a ρ on H is
called a state on H. The set of states on H is denoted by S(H).
Proposition 7.2. The set of states S(H) is convex. The extremal points of
S(H) are the pure states ρ = |uihu| .
Proof. The convexity of the set S(H) is obvious, let us characterize the extremal points of this set. Recall that any state ρ on H admits a canonical
decomposition
X
ρ=
λn |φn ihφn | ,
n∈N
7 QUANTUM PROBABILITY
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P
where the λn are positive with n λn = 1 and the φn ’s form an orthonormal
basis of H. From this representation it is clear that if ρ is extremal then
it is of the form |uihu|. Conversely, take ρ = |uihu| and assume that ρ =
λρ1 + (1 − λ)ρ2 is a convex combination of two states. Put P = I − |uihu|, so
that Tr (ρP
P) = 0 and thus Tr (ρ1 P) = Tr (ρ2 P) = 0. If for every i = 1, 2 we
put ρi = j λij |uij ihuij |, we then get
Tr (ρi P) =
X
λij huij , Puij i = 0 ,
j
P
that is, j λij |huij , ui|2 = 1. The only possibility is that only one of λij is not
null (and thus equal to 1) and the corresponding uij is equal to u. This gives
ρ1 = ρ2 . t
u
Definition 7.3. An observable on H is a self-adjoint operator on H. The
space of observables on H is denoted by O(H).
Let A be an observable on H. By the von Neumann Spectral Theorem
there exists a spectral measure ξ associated to A such that
Z
A=
λ dξ(λ) .
R
If ρ is a state on H then the mapping
µ : Bor(R) −→
[0, 1]
E
7−→ Tr (ρ ξ(E))
is a probability measure on R. This probability measure µ is called the law,
or distribution of A under the state ρ.
The following proposition gives another characterization of the distribution
µ of A.
Proposition 7.4. Let (H, ρ) be a quantum probability space and A be an
observable on H. Then the law of A under the state ρ is the only probability
measure µ on R such that, for all bounded Borel function f on R we have
Z
f (x) dµ(x) = Tr ρ f (A) .
(7.1)
R
Proof. Let µ be the law of A. We write
X
ρ=
λn |un ihun |
n∈N
and as a consequence
X
Tr ρ f (A) =
λn hun , f (A) un i
n∈N
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Stéphane ATTAL
where the series is absolutely convergent. Let ξ denote the spectral measure of
A and let µn be the measure given by µn (E) = hun , ξ(E) un i = kξ(E) un k2 ,
for all n. We then have
Z
X
Tr ρ f (A) =
λn
f (x) dµn (x) .
n∈N
R
On the other hand, if µ is the law of A we have
X
X
µ(E) =
λn hun , ξ(E) un i =
λn µn (E) ,
n∈N
n∈N
P
that is, µ is the measure
n∈N λn µn . This proves that the relation (7.1)
holds when µ is the law of A.
Conversely, if Equation (7.1) holds for a measure µ, then automatically
µ(E) = Tr (ρ 1lE (A)) = Tr (ρ ξ(E)) ,
for all E ∈ Bor(R). This says exactly that µ is the law of A. t
u
Another nice characterization is obtained in terms of the Fourier transform.
Theorem 7.5. Let (H, ρ) be a quantum probability space and let A be an
observable on H. Then the function
f : R −→
R
t 7−→ Tr (ρ eitA )
is the Fourier transform of some probability measure µ on R. This measure
is the law of A under the state ρ.
Proof. For any λ1 , . . . , λn ∈ C and t1 , . . . , tn ∈ R we have
n
X
λk λj f (tk − tj ) =
k,j=1
=
n
X
k,j=1
n
X
λk λj Tr ρ ei(tk −tj )A
λk λj Tr ρ eitk A eitj A
∗ k,j=1

= Tr ρ
n
X
k=1
∗ 
! n
X
λk eitk A 
λj eitj w A   .
j=1
As ρ is positive, the quantity above is positive. By Bochner’s criterion this
proves that f is the Fourier transform of some probability measure µ on R.
By Proposition 7.4 and by uniqueness of the Fourier transform, this measure
is the law of A. t
u
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We now make a remark concerning the action of unitary conjugation in
this setup.
Proposition 7.6. Let ρ be a state and A be an observable on H. If U is a
unitary operator on H, then U∗ A U is still an observable and U ρ U∗ is still
a state. Furthermore, the law of U∗ A U under the state ρ is the same as the
law of A under the state U ρ U∗ .
Proof. The facts that U∗ A U is still an observable and U ρ U∗ is still a state
are obvious and left to the reader. We have
Tr ρ U∗ A U = Tr U ρ U∗ A) .
Furthermore, for any bounded function f on R we have, by the functional
calculus
f (U∗ A U) = U∗ f (A) U ,
which gives finally
Tr (ρ f (U∗ A U)) = Tr (U ρ U∗ f (A)) .
By Proposition 7.4, this says exactly that the law of U∗ A U under the state
ρ is the same as the law of A under the state U ρ U∗ . t
u
7.1.2 Observables vs Random Variables
We stop for a moment our list of definitions and properties and we focus on
the relations between observables and usual random variables in Probability
Theory.
For the simplicity of the discussion we consider only the case of pure states,
the general case can be easily deduced by a procedure called the G.N.S.
representation, which is not worth developing for this discussion.
A Hilbert space H and a wave function ϕ being given, let us see that any
observable on H can be viewed as a classical random variable. Let A be an
observable on H, i.e. a self-adjoint operator on H. Consider the probability
measure
2
E 7→ µ(E) = k1lE (A) ϕk
on (R, Bor(R)). Consider the operator U from L2 (R, Bor(R), µ) to H given
by Uf = f (A) ϕ. This is a well-defined operator and an isometry, for the von
Neumann Spectral Theorem gives
Z
2
2
kf (A) ϕk =
|f (x)| dµ(x) .
R
We deduce easily from this definition
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Stéphane ATTAL
U∗ f (A) ϕ = f
and
U∗ ϕ = 1l .
The operator U∗ A U is then clearly equal to the operator MX of multiplication by the function X(x) = x on R. Furthermore, we have
h ϕ , f (A) ϕ i = h U∗ ϕ , U∗ f (A)U U∗ ϕ i
= h 1l , Mf (X) 1l i
Z
=
f (X(x)) dµ(x) .
R
This means that the probability distribution associated to the observable A,
in the state ϕ, as described in the setup of Quantum Probability Theory, is
the same as the usual probability distribution of X as a random variable on
(R, Bor(R), µ).
Being given a single observable A on H together with a state ϕ is thus the
same as being given a classical random variable X on some probability space
(Ω, F, P). The observable A specifies the random variable X as a function on
Ω, the state ϕ specifies the underlying probability measure P and therefore
specifies the probability distribution of X.
In the converse direction, it is easy to see a classical probabilistic setup
as a particular case of the quantum setup. Let X be a real random variable
defined on some probability space (Ω, F, P). Put H = L2 (Ω, F, P) and ϕ = 1l.
The operator A = MX is then a self-adjoint operator on H. Let us compute
the probability distribution of the observable A in the state ϕ, in the sense
of Quantum Probability. We get
h ϕ , f (A) ϕ i = h 1l , Mf (X) 1l i
Z
=
f (X(ω)) dP(ω)
R
= E[f (X)] .
Thus, the distribution of the observable A, in the quantum sense is the same
as the probability distribution of the random variable X in the usual sense.
One could be tempted to conclude that there is no difference between
the probabilistic formalism of Quantum Mechanics and the usual Probability
Theory. But this is only true for one observable, or actually for a family
of commuting observables. Indeed, once considering two observables A and
B on H which do not commute, the picture is completely different. Each of
the observables A and B can be seen as concrete random variables X and Y
respectively, but on some different probability spaces. Indeed, if A and
B do not commute they cannot be diagonalized simultaneously and they are
7 QUANTUM PROBABILITY
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represented as multiplication operators on some different probability spaces
(otherwise they would commute!).
Furthermore, their associated probability spaces have nothing to do together. They cannot be put together, as one usually does in the case of independent random variables, by taking the tensor product of the two spaces,
because the two random variables X and Y are not independent. As operators on H the observables A and B may have some strong relations, such
as [A, B] = λI, for example (this is the case for momentum and position
observables in Quantum Mechanics).
Is there a way, with the random variables X, Y and their associated probability spaces, to give an account of such a relation? Aspect’s experiment and
Bell’s inequalities prove that this is actually impossible! As an example, the
spin of an electron in two different directions gives rise to two Bernoulli random variables but each one on its own probability space. These two Bernoulli
random variables are not at all independent, they depend of each other in a
way which is impossible to express in classical terms. The only way to express
their dependency is to represent them as 2 × 2 spin matrices on C2 and work
with the quantum mechanical axioms!
As a conclusion, each single observable in Quantum Mechanics is like a
concrete random variable, but on its own probability space, “with its own
dice”. Two non-commuting observables have different associated probability
spaces, “different dices”; they cannot be represented by independent random
variables or anything else (even more complicated) in the classical language of
probability. The only way to express their mutual dependency is to consider
them as two self-adjoint operators on the same Hilbert space and to compute
their probability distributions as the quantum mechanical axioms dictate us.
In the quantum formalism it is possible to make operations on non commuting observables A, B, for example to add them and to consider the result
as a new observable. The distribution of A + B has then nothing to do with
the convolution of the distribution of A by the one of B, it depends in a complicated way on the relations between A and B as operators on H. We shall
see examples later on in this course.
The case of commuting observables is equivalent to usual case of a pair of
random variables. Let us here give some details about that.
Let A and B be two observables on H which commute (in the case of unbounded operators this means that their spectral measures commute). Then
there exists a spectral measure dξ(x, y) on R × R such that
Z
Z
A=
x dξ(x, y) and B =
y dξ(x, y) .
R2
R2
In other words, A and B can be diagonalized simultaneously, they can be
represented as multiplication operators on the same probability space.
The pair (A, B) admits a law in the quantum sense. Indeed, if ρ is a state
on H, then the mapping
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Stéphane ATTAL
µ : Bor(R × R) −→
[0, 1]
E×F
7−→ Tr ρ ξ(E, F )
extends to a probability measure on R × R. This probability measure is called
the law of the pair (A, B) under the state ρ. Another way to understand this
law is to say that if A and B commute they admit a two variable functional
calculus and, for every bounded measurable function f on R × R, we have
Z
f (x, y) dµ(x, y) = Tr (ρ f (A, B)) .
R2
From the Fourier transform point of view, the law µ is obtained by noticing
that if A and B commute then the function
(t1 , t2 ) 7−→ Tr ρ ei(t1 A+t2 B)
satisfies Bochner’s criterion for two variable functions and hence is the Fourier
transform of some probability measure µ on R × R. Clearly, this measure is
the law of (A, B) under ρ as defined above.
At this stage, it is interesting to note a way to produce independent random
variables in the quantum probability context.
Proposition 7.7. Let A and B be two observables on H and K respectively.
Let µ and ν denote their laws under the states ρ and τ , respectively. On the
Hilbert space H ⊗ K consider the commuting observables
b =A⊗I
A
b = I ⊗ B.
B
and
b B)
b follows the law
Then, under the state ρ ⊗ τ , the pair of observables (A,
µ ⊗ ν, that is, they are independent observables with the same individual law
as A and B respectively.
b and B
b commute, hence they admit a law of pair, let us
Proof. Clearly A
compute this law. For every bounded real functions f and g we have
b = f (A) ⊗ I
f (A)
b = I ⊗ g(B) .
g(B)
and
Thus
b B)
b = Tr ρ f (A) ⊗ τ g(B)
Tr (ρ ⊗ τ ) f (A)g(
= Tr ρ f (A) Tr τ g(B)
Z
Z
=
f (x) dµ(x)
g(y) dν(y)
R
ZR Z
=
f (x)g(y) dµ(x) dν(y) .
R
R
7 QUANTUM PROBABILITY
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b has the same law under ρ ⊗ τ
Taking g = 1l, the identity above shows that A
as the one of A under the state ρ. The corresponding property of course holds
for B. Taking general f and g, the identity above shows the independence of
b and B.
b t
A
u
7.1.3 The Role of Multiplication Operators
Together with the discussion we had above on the connection between observables and classical random variables, I want to put the emphasis here on
the particular role of multiplication operators in the quantum setup.
Consider a probability space (Ω, F, P) and a family of real random variables on (Xi )i∈I which are of interest for some reasons. Imagine we have
found a natural unitary isomorphism U : L2 (Ω, F, P) → H with some Hilbert
space H. The question we want to discuss here is the following: Where can
we read on H the random variables Xi , with all their probabilistic properties
(individual distributions, relations with each other such as independence or
dependence, joint distributions, functional calculus, etc.)?
This is certainly not by looking at the images hi = UXi of the Xi ’s in H.
Indeed, these elements hi of H could be almost any element of H by choosing
correctly the unitary operator U. They carry no probabilistic information
whatsoever on the random variables Xi .
The pertinent objects to look at are the operators
Ai = U MXi U∗ ,
that is, the push-forward of the operators of multiplications by the Xi ’s.
Indeed, these operators are a commuting family of self-adjoint operators on
H. The functional calculus for such families of operators gives easily
f (Ai1 , . . . , Ain ) = U Mf (Xi1 ,...,Xin ) U∗ .
If we put φ = U1l, then φ is a pure state of H and we have
Tr (|φihφ| f (Ai1 , . . . , Ain )) = hφ , f (Ai1 , . . . , Ain )φi
= h1l , U∗ f (Ai1 , . . . , Ain ) U 1li
D
E
= 1l , Mf (Xi1 ,...,Xin ) 1l
= E [f (Xi1 , . . . , Xin )] .
The law of the n-uplet (Ai1 , . . . , Ain ) as observables in the state |φihφ| is thus
the same as the usual law of the n-uplet (Xi1 , . . . , Xin ).
The operators Ai contain all the probabilistic informations on the Xi ’s:
individual and joint laws, functional calculus, etc. The operators Ai play
exactly the same role in H as the random variables Xi play in L2 (Ω, F, P).
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Stéphane ATTAL
The only informations concerning the Xi ’s that one cannot recover from
the Ai ’s are the samples, that is, the individual values Xi (ω) for ω ∈ Ω.
We end this discussion with an extension to a situation which is very
common in Quantum Noise Theory. Together with the situation described
above, we have another Hilbert space K and an operator on K ⊗ H of the
form
n
X
T=
Bi ⊗ Ai ,
i=1
for some bounded operators Bi on K. On the other hand, consider the random
operator
S : Ω −→
B(K)
P
n
ω 7−→ S(ω) = i=1 Xi (ω)Bi .
We want to make clear the connections between T and S and show that they
are one and the same thing.
Let us define the “multiplication” operator associated to S:
b
S : L2 (Ω, F, P) ⊗ H −→ P
L2 (Ω, F, P) ⊗ H
n
f ⊗h
7−→ i=1 Xi f ⊗ Bi h .
In other words
b
S=
n
X
MXi ⊗ Bi .
i=1
Then the connection becomes clear:
T = (U ⊗ I) b
S (U∗ ⊗ I) .
Even more than just this unitary equivalence, the operator T contains all
the probabilistic informations of the random operator S; any kind of computation that one wants to perform on S can be made in the same way on
T. The functional calculus is the same, the expectations and the laws are
computed is the same way as described above.
7.1.4 Events
In this subsection we discuss the notion of event in Quantum Probability.
In classical probability theory an event E can be identified to the random
variable X = 1lE . That is, a random variable X such that X 2 = X = X.
Conversely, every random variable X which satisfies the above relation is
the indicator function of some event. On the space L2 (Ω, F, P), the indicator
function X = 1lE can be identified with its multiplication operator f 7→ 1lE f ,
7 QUANTUM PROBABILITY
11
it is thus an orthogonal projection of L2 (Ω, F, P), whose range is the subspace
of functions with support included in E.
Definition 7.8. Following the analogy with the classical case, we define an
event on a quantum probability space (H, ρ) to be any orthogonal projector
on H. We denote by P(H) the set of events of H. In particular, if A is an
observable on H, with spectral measure ξA and if E is a Borel set of R, the
projector ξA (E) can be interpreted as the event “A belongs to E”. Up to a
normalization constant it is the state one would obtain after a measurement of
the observable A, if one had observed “the value of the measurement belongs
to the set E”. In other words, the subspace Ran ξA (E) is the subspace of wave
functions for which a measure of A would give a result in E with probability
1; this subspace is indeed the subspace corresponding to the knowledge “A
belongs to E ” on the system.
This terminology coincides also with the definitions and results of previous
subsections, for by Proposition 7.4 we have
Z
Tr (ρ ξA (E)) = Tr (ρ 1lE (A)) =
1lE (x) dµ(x) = µ(E)
R
which says that Tr (ρ ξA (E)) is the probability that A belongs to E under the
state ρ.
Definition 7.9. Together with this quantum notion of event we make use
the following terminology:
– If P1 , P2 are two events and if P1 ≤ P2 , we say that P1 implies P2 .
– The operators 0 and I are the null and certain events.
– The complementary of the event P is the event I − P.
– If P1 , . . . , Pn are events then ∨ni=1 Pi is the orthogonal projector onto the
subspace generated by ∪ni=1 Ran Pi ; it is the event “occurrence of at least
one of the Pi ’s”. In the same way ∧ni=1 Pi is the orthogonal projector onto
∩ni=1 Ran Pi ; it is the event “simultaneous occurrence of all the Pi ’s”.
Definition 7.10. When a state ρ is given, the mapping α : P 7→ Tr (ρ P)
behaves like a probability measure on P(H) in the sense that
i) α(I) = 1
P
P
ii) α
i∈N Pi =
i∈N α(Pi ), for any family (Pi )i∈N of pairwise orthogonal
projections.
A mapping α : P(H) → R which satisfies i) and ii) is called a probability
measure on P(H). Note that every probability measure α on P(H) satisfies:
iii) α(P) ∈ [0, 1] for all P ∈ P(H).
iv) α(I − P) = 1 − α(P) for all P ∈ P(H).
v) α(0) = 0.
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Stéphane ATTAL
One may naturally wonder what is the general form of all probability
measures on P(H). The answer is given by the celebrated Gleason’s theorem.
This theorem is very deep and has many applications, developments and
consequences in the literature. Its proof is very long and heavy. As we do not
really need it in the following (we just present it as a remark), we state it
without proof. One may consult the book [Par92], pp. 31–40, for a complete
proof.
Theorem 7.11 (Gleason’s theorem). Let H be a separable Hilbert space with
dimension greater than or equal to 3. A mapping α : P(H) → R is a probability measure on P(H) if and only if there exists a density matrix ρ on H
such that
α(P) = Tr (ρ P)
for all P ∈ P(H).
The correspondence ρ 7→ α between density matrices on H and probability
measures on P(H) is a bijection.
7.2 Quantum Bernoulli
We shall now enter into concrete examples of Quantum Probability spaces and
specific properties of this quantum setup. In this section we aim to explore
the simplest situation possible, the quantum analogue of a Bernoulli random
walk. We shall see that this simple situation in classical probability becomes
incredibly richer in the quantum context.
7.2.1 Quantum Bernoulli Random Variables
Before going to the Bernoulli random walks, we study the quantum analogue
of Bernoulli random variables. This is to say that we study now the simplest
non-trivial example of a quantum probability space: the space H = C2 .
Definition 7.12. Let (e1 , e2 ) be the canonical basis of C2 . The matrices
01
0 −i
1 0
σ1 =
, σ2 =
, σ3 =
10
i 0
0 −1
are the so-called Pauli matrices. There are also usually denoted by σx , σy , σz ,
respectively.
With the help of these matrices we have very useful parametrizations of
observables and states.
7 QUANTUM PROBABILITY
13
Proposition 7.13.
1) Together with the identity matrix I, the Pauli matrices form a real basis
of the space O(H). That is, any observable A on H can be written
A = tI +
3
X
xi σi
i=1
for some t, x1 , x2 , x3 ∈ R. In particular we get Tr (A) = 2t.
2) Putting
q
x21 + x22 + x23 ,
the eigenvalues of A are t − kxk , t + kxk .
kxk =
3) Any state ρ is of the form
ρ=
1
2
1 + x y − iz
y + iz 1 − x
with x, y, z ∈ R satisfying x2 + y 2 + z 2 ≤ 1. In particular, ρ is a pure state if
and only if x2 + y 2 + z 2 = 1.
Proof. 1) and 2) are immediate and left to the reader. Let us prove 3). A
state ρ on H is an observable which has trace 1 and positive eingenvalues.
Hence by 1) and 2), a state ρ on H is of the form
!
3
X
1
xi σi
ρ=
I+
2
i=1
with kxk ≤ 1. As a consequence the space S(H) identifies to B(0, 1), the
closed unit ball of R3 , with the same convex structure. As a consequence, the
extreme points of S(H), the pure states, correspond to the extreme points of
B(0, 1), that is, the unit sphere S 2 of R3 . t
u
The result 3) above means that one can always write a pure state as:


1 + cos(ϕ) e−iθ sin(ϕ)
1

ρ= 
2
eiθ sin(ϕ) 1 − cos(ϕ)

=
cos2
eiθ sin
ϕ
2
ϕ
2
e−iθ sin
cos
ϕ
2
ϕ
2
sin2
cos
ϕ
2
ϕ
2

,
for some θ ∈ [0, 2π] and ϕ ∈ [0, π]. We recognize the operator |uihu| where
14
Stéphane ATTAL
θ
u=
e−i 2 cos
θ
ei 2 sin
!
ϕ
2
ϕ
2
.
This way, one cleary describes all unitary vectors of C2 up to a phase factor,
that is, one describes all rank one projectors |uihu|.
We shall now compute the law of quantum observables in this context.
Proposition 7.14. Consider an orthonormal basis of C2 for which the state
ρ is diagonal:
p 0
ρ=
,
0q
with 0 ≤ q ≤ p ≤ 1 and p + q = 1. If A is any observable
A = tI +
3
X
xi σi ,
i=1
then the law of A under the state ρ is the law of a random variable which
takes the values t − kxk and t + kxk with probability
x3
1
(q − p)
1+
2
kxk
and
1
x3
(q − p)
1−
2
kxk
respectively.
Proof. The eigenvalues of A are t − kxk and t + kxk, as we have already seen.
The eigenvectors are easily obtained too, they are of the form
−x1 + ix2
x1 − ix2
v1 = λ
v2 = µ
,
kxk + x3
kxk − x3
λ, µ ∈ C, respectively.
Taking the normalized version u1 , u2 of these vectors, the probability of
measuring A to be equal to t − kxk is then
1
p(x21 + x22 ) + q(kxk + x3 )2
2 kxk (kxk + x3 )
p(kxk − x3 ) + q(kxk + x3 )
=
2 kxk
1
x3
=
1+
(q − p) .
2
kxk
hu1 , ρ u1 i =
This also gives immediately the other probability. t
u
In the particular case p = 1 and q = 0, i.e. if ρ = |e1 ihe1 |, we get all
probability distribution on R with (at most) 2 point support. Note that this
7 QUANTUM PROBABILITY
15
situation would be impossible in a classical probability context. Indeed, let
Ω = {0, 1} and P be a fixed probability measure on Ω with P(1) = p and
P(0) = q. Then a random variable X on Ω can take any couple of values (x, y)
in R but only with P(X = x) = p and P(X = y) = q. In quantum probability,
we observe that, under a fixed state ρ, the set of observables gives rise to the
whole range of two-point support laws. The space of observables is much
richer than in the classical context.
We end up this subsection with an illustration of non-commutativity in this
context. Let ρ = |e1 ihe1 |. The observables σ1 , σ2 both have the distribution
1
1
2 δ−1 + 2 δ1 under that state. But the observable
0 1−i
σ1 + σ2 =
1+i 0
√ √
has spectrum {− 2, 2} and distribution
1
1 √
δ
+ δ√ .
2 − 2 2 2
Adding two Bernoulli random variables with values {−1, √
+1},√we have ended
up with a Bernoulli random variable with values in {− 2, 2} ! This fact
would be of course impossible in the classical context, it comes from the fact
that σ1 and σ2 do not commute, they do not admit a pair law. Let us check
this fact explicitely. When computing
ϕ(t1 , t2 ) = he1 , ei(t1 σ1 +t2 σ2 ) e1 i
p
one easily finds ϕ(t1 , t2 ) = cos
t21 + t22 which is not the Fourier transform
of a positive measure on R2 . Indeed, Bochner’s positivity criterion
n
X
λi λ̄j ϕ(ti − tj , si − sj ) ≥ 0
i,j=1
fails when taking n = 3, λ1 = λ2 = λ3 = 1, t1 = s2 = t3 = s3 = 2π/3 and
s1 = t2 = 0.
7.2.2 The Toy Fock Space
In this subsection we go beyond the simple example studied above and we
focus on the very rich and fundamental structure of a chain of quantum
Bernoulli random variables. The constructions and results of this subsection
are fundamental in discrete time Quantum Noise Theory.
16
Stéphane ATTAL
Consider the space C2 with a fixed orthonormal basis which we shall denote
by {χ0 , χ1 }. Physically, one can think of this basis as representing a two-level
system with its ground state χ0 and excited state χ1 , or a spin state space
with spin down χ0 and spin up χ1 states etc.
Definition 7.15. We define the Toy Fock space 1 to be the countable tensor
product
O
C2 ,
TΦ =
N∗
associated to the stabilizing sequence (χ0 )n∈N∗ . In other words, it is the
Hilbert space whose Hilbertian orthonormal basis is given by the elements
e1 ⊗ e2 ⊗ · · · ⊗ en ⊗ · · ·
where for each i we have ei = χ0 or χ1 , but only a finite number of ei are
equal to χ1 .
Definition 7.16. Let us describe the above orthonormal basis in another
useful way. Let
P = Pf (N∗ )
be the set of finite subsets of N∗ . An element σ of P is thus a finite sequence
of integers σ = {i1 < . . . < in }. For every σ = {i1 < . . . < in } ∈ P we put
χσ to be the basis element e1 ⊗ · · · ⊗ en ⊗ · · · of TΦ given by
(
χ1 if j = ik for some k,
ej =
χ0 otherwise.
Note the particular case
χ∅ = χ0 ⊗ . . . ⊗ χ0 ⊗ . . .
Every element f of TΦ writes in unique way as
X
f=
f (σ) χσ ,
σ∈P
with
2
kf kTΦ =
X
2
|f (σ)| < ∞ .
σ∈P
In other words, with these notations, the space TΦ naturally identifies to the
space `2 (P).
The fundamental structure of TΦ is the one of a countable tensor product
of copies of C2 with respect to a given basis vector χ0 . Actually a more
1
The toy Fock space was called “bébé Fock” in french, by P.-A. Meyer. Exercise:
find the joke.
7 QUANTUM PROBABILITY
17
abstract definition for TΦ is possible as follows: TΦ is the separable Hilbert
space whose orthonormal basis is fixed and chosen to be indexed by P, the
set of finite subsets of N∗ . This is enough to carry all the properties we need
for the space TΦ.
Definition 7.17. Let n ≤ m be any fixed elements of N∗ . We denote by
TΦ[n,m] the subspace of TΦ generated by the χσ such that σ P
⊂ [n, m]. The
space TΦ[n,m] is thus the subspace of TΦ made of those f = σ∈P f (σ)χσ
such that f (σ) = 0 once σ 6⊂ [n, m].
Among those spaces TΦ[n,m] , we denote by TΦn] the space TΦ[1,n] . We
also write TΦ[n for the space TΦ[n,+∞[ , with obvious definition.
Following the same idea, we denote by P[n,m] the set of finite subsets of
N∗ ∩ [n, m] and we have the corresponding notations Pn] and P[n . We then
get the natural identification
TΦ[n,m] = `2 (P[n,m] ) .
For a σ ∈ P, we put σ[n,m] = σ ∩ [n, m]. In the same way we define σn] =
σ ∩ [1, n] and σ[n = σ ∩ [n, +∞[.
With these notations, the fundamental structure of TΦ is reflected by the
following result.
Proposition 7.18. Let n < m ∈ N∗ be fixed. The mapping
U : TΦn] ⊗ TΦ[n+1,m−1] ⊗ TΦ[m −→ TΦ
f ⊗g⊗h
7−→ k
where
k(σ) = f (σn] ) g(σ[n+1,m−1] ) h(σ[m ) ,
for all σ ∈ P, extends to a unitary operator.
Proof. The operator U is isometric for
hU(f ⊗ g ⊗ h) , U(f 0 ⊗ g 0 ⊗ h0 )iTΦ =
X
f (σn] ) g(σ[n+1,m−1] ) h(σ[m ) f 0 (σn] ) g 0 (σ[n+1,m−1] ) h0 (σ[m )
=
σ∈P
=
X
X
f (α) f 0 (α)
α∈Pn]
g(β) g 0 (β)
0
h(γ) h0 (γ)
γ∈P[m
β∈P[n+1,m−1]
0
X
0
= hf , f iTΦn] hg , g iTΦ[n+1,m−1] hh , h iTΦ[m
= hf ⊗ g ⊗ h , f 0 ⊗ g 0 ⊗ h0 iTΦn] ⊗TΦ[n+1,m−1] ⊗TΦ[m .
But U is defined on a dense subspace and thus extends to an isometry on the
whole of TΦn] ⊗ TΦ[n−1,m+1] ⊗ TΦ[m . Its range is dense in TΦ for it contains
all the χσ ’s. Thus U is unitary. t
u
18
Stéphane ATTAL
We have completely described the structure of the space TΦ. We now turn
to the operators on TΦ.
Definition 7.19. We shall make use of a particular basis, which is not the
usual Pauli matrix basis, as in the previous subsection. In the orthonormal
basis {χ0 , χ1 } of C2 we consider the basis of matrices
a00
=
10
00
01
00
0
1
1
, a1 =
, a0 =
, a1 =
.
00
10
00
01
In other words, we have
aij χk = δk,i χj
for all i, j, k = 0, 1.
From these basic operators on C2 , we built basic operators on TΦ by
considering the operators aij (n) which act as aij on the n-th copy of C2 and
as the identity on the other copies. This means, when acting on the natural
orthonormal basis of TΦ
a00 (n)χσ = 1ln6∈σ χσ
a01 (n)χσ = 1ln6∈σ χσ∪{n}
a10 (n)χσ = 1ln∈σ χσ\{n}
a11 (n)χσ = 1ln∈σ χσ .
In some sense the operators aij (n) form a basis of B(H). Of course, one
needs to be precise with the meaning of such a sentence, for now we deal
with an infinite dimensional space. This could be done easily in terms of von
Neumann algebras: “The von Neumann algebra generated by the operators
aij (n), i, j = 0, 1, n ∈ N∗ , is the whole of B(TΦ).” But this would bring us
too far and would need some knowledge on operator algebras which is not
required for this lecture.
Let us describe how random walks can be produced with the help of the
operators aij (n).
Theorem 7.20. Let A be an observable on C2 , with coefficients
X
A=
αji aij .
i,j=0,1
The observable A admits a certain law µ under the state χ0 . For all n ∈ N∗
define the observable
X
A(n) =
αji aij (n)
i,j=0,1
7 QUANTUM PROBABILITY
19
on TΦ. Then the sequence (A(n))n∈N in B(TΦ) is commutative and, under
the state χ∅ , it has the law of a sequence of independent random variables,
each of which has the same law µ.
Proof. This is an easy application of Proposition 7.7 and Proposition 7.18
when noticing that:
i) for every n ∈ N∗ , in the following splitting of TΦ:
TΦ ' TΦn−1] ⊗ TΦ[n,n] ⊗ TΦ[n+1
the operator A(n) is of the form I ⊗ A(n) ⊗ I.
ii) the state χ∅ on TΦ is the tensor product ⊗n∈N∗ Ω of the ground states of
each copy of C2 .
Details are left to the reader. t
u
Of course in the result above one can easily make the αji ’s depend on n
also; this allows to produce any sequence of independent random variables.
Summing up these random variables, this provides an easy way of realizing
any classical random walk, in law, on TΦ. We shall see in next subsection that
the Toy Fock space provides much more than that: it can realize the multiplication operators of any classical Bernoulli random walk on its canonical
space, by means of linear combinations of the operators aij (n).
7.2.3 Probabilistic Interpretations
In this section we explore the so-called probabilistic interpretations of TΦ,
that is, we shall realize on TΦ the multiplication operators associated to any
Bernoulli random walk, by means of only linear combinations of the operators
aij (n).
Definition 7.21. Let p ∈]0, 1[ be fixed and q = 1 − p. Consider a classical
Bernoulli sequence (νn )n∈N∗ , that is, a sequence of independent identically
distributed Bernoulli random variables νn with law pδ1 + qδ0 . We realize
∗
this random walk on its canonical space (Ω, F, µp ) where Ω = {0, 1}N , F is
the σ-field generated by finite-based cylinders and µp is the only probability
measure on (Ω, F) which makes the coordinate mappings
νn : Ω −→ {0, 1}
ω 7−→ ωn
being independent, identically distributed with law pδ1 + qδ0 .
Definition 7.22. We center and normalize νn by putting
νn − p
Xn = √
pq
20
Stéphane ATTAL
so that E[Xn ] = 0 and E[Xn2 ] = 1. The random variables Xn are then independent, they take the values
q
 q,
with probability p ,
p
q
p
−
q , with probability q .
For every σ ∈ P we put
(
Xi1 · · · Xin
Xσ =
1l
if σ = {i1 , . . . , in } ,
if σ = ∅ ,
where 1l is the deterministic random variable always equal to 1.
Proposition 7.23. The set {Xσ ; σ∈P} forms an orthonormal basis of the
Hilbert space L2 (Ω, F, µp ).
Proof. Let us first check that the set {Xσ ; σ∈P} forms an orthonormal set.
Let α, β ∈ P be fixed, we have
hXα , Xβ iL2 (Ω,F ,µp ) = Eµp [Xα Xβ ]
= Eµp [Xα\β (Xα∩β )2 Xβ\α ]
2
= Eµp [Xα\β ] Eµp [Xα∩β
] Eµp [Xβ\α ] ,
for the random variables Xi and Xj are independent once i 6= j. As we have
E[Xσ ] = 0 for all σ 6= ∅, we get that hXα , Xβ iL2 (Ω,F ,µp ) vanishes unless
α \ β = β \ α = ∅, that is, unless α = β. In the case where α = β we get
hXα , Xβ iL2 (Ω,F ,µp ) = Eµp [Xσ2 ] = 1 .
This proves the orthonormality.
Let us now prove the totality of the Xσ ’s. Had we replaced N∗ by
{1, . . . , N } in the above definition of Ω, we would directly conclude that
the Xσ form a basis for they are orthonormal and have cardinal 2N , the dimension of the space L2 (Ω, F, µp ) in that case. We conclude in the general
∗
case Ω = {0, 1}N by noticing that any f ∈ L2 (Ω, F, µp ) can be approached
by finitely supported functions. t
u
From this proposition we see that there is a natural isomorphism between
L2 (Ω, F, µp ) and the toy Fock space TΦ: it consists in identifying the orthonormal basis {Xσ , σ ∈ P} of L2 (Ω, F, µp ) with the orthonormal basis
{χσ , σ ∈ P} of TΦ. The space L2 (Ω, F, µp ) is now denoted by TΦp and
is called the p-probabilistic interpretation of TΦ. The space TΦp is an exact
reproduction of TΦ, but it is concretely represented as the space of L2 functionals of some random walk. The element Xσ of TΦ interprets in TΦp as a
concrete random variable Xσ .
7 QUANTUM PROBABILITY
21
All the spaces TΦp identify pairwise this way and they all identify to TΦ.
But this identification only belongs to the level of Hilbert spaces. As we have
already discussed in Subsection 7.1.3, the fact that the random variable Xn in
the p-probabilistic interpretation TΦp is represented by the basis vector χ{n}
in TΦ does not mean much. All the probabilistic properties of Xn such as its
law, its independence with respect to the other Xm ’s, ... all these informations are lost in this identification. The actual representation of the random
variable Xn of TΦp in TΦ should be the push forward of the multiplication
operator by Xn .
Let us be more precise.
Definition 7.24. Let Up : TΦp 7−→ TΦ be the unitary operator which realizes the basis identification between TΦp and TΦ. Let MXn be the operator
of multiplication by Xn in TΦp . We shall consider the operator
MpXn = Up MXn U∗p
on TΦ, which is the representation of the random variable Xn but in the
space TΦ.
We shall now prove a striking result which shows that these different random variables Xn , for different p’s, are represented on TΦ by means of a very
simple linear combination of the aij (n)’s.
Theorem 7.25. For all p ∈]0, 1[ and every n ∈ N∗ , the operator MpXn on
TΦ is given by
MpXn = a01 (n) + a10 (n) + cp a11 (n) ,
(7.2)
where
q−p
cp = √ .
pq
The mapping p 7→ cp is a bijection from ]0, 1[ to R.
Proof. Consider a f ∈ TΦp , the product Xn f is clearly determined by the
products Xn Xσ , σ ∈ P. If n does not belong to σ then by definition of our
basis we get
Xn Xσ = Xσ∪{n} ,
that is, there is nothing to compute. If n belongs to σ then
Xn Xσ = Xn2 Xσ\{n} .
In other words, the product Xn f on TΦp is determined by the Xn2 , n ∈ N∗ .
Lemma 7.26. On TΦp we have, for all n ∈ N∗
22
Stéphane ATTAL
Xn2 = 1l + cp Xn
where
(7.3)
q−p
cp = √ .
pq
Proof (of the lemma). We have on TΦp
2
νn − p
ν 2 − 2pνn + p2
= n
√
pq
pq
p
(1 − 2p)νn
+
=
pq
q
p
q−p √
=
pqXn + p1l + 1l
pq
q
q−p
= 1l + √ Xn .
pq
Xn2 =
This proves (7.3) and the lemma.
Coming back to the proof of the theorem and applying Lemma 7.26, we get
for all σ ∈ P
Xn Xσ = 1ln6∈σ Xσ∪{n} + 1ln∈σ (1l + cp Xn ) Xσ\{n}
= 1ln6∈σ Xσ∪{n} + 1ln∈σ Xσ\{n} + cp 1ln∈σ Xσ
This means that on TΦ we have
MpXn Xσ = 1ln6∈σ Xσ∪{n} + 1ln∈σ Xσ\{n} + cp 1ln∈σ Xσ .
We recognize the action of the operator
a01 (n) + a10 (n) + cp a11 (n)
on the basis of TΦ. This gives (7.2).
Finally, we have
1 − 2p
q−p
=p
cp = √
pq
p(1 − p)
and as a function of p it is easy to check that it is a bijection from ]0, 1[ to
R. t
u
What we have seen throughout this section is fundamental. On a very
simple space TΦ we have been able to reproduce a continuum of different
probabilistic situation: all the Bernoulli random walk with any parameter
p ∈]0, 1[. We are able to completely reproduce inside TΦ all these different
probability spaces, all these different laws. Even more surprising is the fact
that this representation is achieved only with the help of linear combinations
7 QUANTUM PROBABILITY
23
of three basic processes (a10 (n))n∈N∗ , (a01 (n))n∈N∗ , (a11 (n))n∈N∗ . These three
quantum processes play the role of three fundamental quantum Bernoulli
random walk from which every classical Bernoulli random walk can be constructed.
7.2.4 Brownian Motion and Poisson Process
The fact that one gets all classical Bernoulli random variables from the Toy
Fock space has many important consequences, some of them are fundamental
in the theory of quantum noises. We shall give here a taste of it by showing
that the toy Fock space gives rise to two fundamental stochastic processes:
the Brownian motion and the Poisson process.
First of all, as we wish to give some heuristic of a continuous-time limit, we
shall now have our Toy Fock indexed by hN∗ , for some small real parameter
h > 0, instead of N∗ .
In Theorem 7.25, consider the case p = 1/2. The random variables Xn
then take the values ±1 with probability 1/2. In that case we get c 21 = 0,
which means that the operators
Snh =
n
X
a01 (ih) + a10 (ih) , n ∈ N∗ ,
i=1
are a commutative sequence of observables on TΦ, their distribution in the
state X∅ is the one of a symmetric Bernoulli random walk indexed by hN∗ .
Even more, they are the multiplication operators by these random walks on
their canonical space. This means in particular that all the functional calculus
that can be usually derived from these random walks, can be obtained in the
same way with the sequence of operators (Snh ).
Renormalizing (Snh ), we consider the sequence
Qnh =
n √
X
h a01 (ih) + a10 (ih) , n ∈ N∗ ,
i=1
which is a random walk converging to a Brownian motion when h tends to
0. This is a convergence in law for the associated random walk, but when
regarding the operators Qnh this is far more. It is a convergence at the level
of multiplication operators, it is a convergence which respects the functional
calculus of the Brownian motion in the limit. For example, computing
24
Stéphane ATTAL
n
n−1
X
X
2
(Qih − Q(i−1)h )2 =
h a01 (ih) + a10 (ih)
i=2
i=1
2
n
X
01
=
h
1 0 ih
i=2
n
X
10
=
h
0 1 ih
i=2
= hI,
gives directly the quadratic variation of the Brownian motion:
(dWt )2 = dt
or more precisely
[W , W ]t = t .
Even the Ito formula is encoded in this matrix representation. For example,
the famous formula
Z t
2
Wt = 2
Ws dWs + t
0
appears as a simple continuous time limit of the following computation
Q2nh =
n
X
h a01 (ih) + a10 (ih)
a01 (jh) + a10 (jh)
i,j=1
=2
n
X


i−1
X

h a01 (jh) + a10 (jh)  a01 (ih) + a10 (ih)
i=2
+
j=1
n
X
2
h a01 (ih) + a10 (ih)
i=1
=2
n
X
Q(i−1)h Q(ih) − Q((i − 1)h) + h I .
i=2
Where the situation becomes even more striking is the way the Poisson
process enters into the game also, in the Toy Fock space.
Coming back to the random walks of Theorem 7.25, but with a particular
choice of the probability p now. Indeed, let us take p = h, where h is the
same parameter as the time step of the random walk. In order to make the
computation not too huggly we prefer to take
p=
so that
h
,
1+h
q=
1
,
1+h
7 QUANTUM PROBABILITY
25
1−h
cp = √ .
h
√
√
The associated random variable Xnh takes the values 1/ h and − h with
probability p and q respectively. The random walk
Snh =
n √
X
h Xih
i=1
is a Bernoulli random walk, with time step h, which takes the value −h very
often (with probability 1 − h, more or less) and the value 1 very rarely (with
probability h, more or less). In the continuous time limit it is a compensated
Poisson process.
From the point of view of the Toy Fock space representation we obtain the
operators
Xnh =
n √
X
h a01 (ih) + a10 (ih) + cp a11 (ih)
i=1
=
n √
X
h a01 (ih) + a10 (ih) + (1 − h)a11 (ih) .
i=1
This operator family is supposed to behave like a compensated Poisson
process, in the limit h → 0. As a consequence, the family
Nnh = Xnh + nhI =
n √
X
h a01 (ih) + a10 (ih) + a11 (ih) + ha00 (ih)
i=1
should represent a standard Poisson process in the limit. In the same way
as for the Brownian motion, one can recover from these operators all the
properties of the Poisson process, of its functional calculus. For example, the
matrix
√ √
h
h
0
1
1
0
A = h a1 (ih) + a0 (ih) + a1 (ih) + ha0 (ih) = √
h 1 (ih)
satisfies
A2 = (1 + h)A .
This relation gives in the continuous time limit the fundamental characterization of the Poisson process
(dNt )2 = dNt ,
or else
[N , N ]t = Nt .
26
Stéphane ATTAL
7.3 Higher Level Chains
The aim of this section is to present the natural extension of the Toy Fock
space structure, but for higher number of levels on each sites. From a quantum
mechanical point of view this is clearly a richer physical structure. From the
probabilistic interpretation point of view it gives access to all random walks
in RN . But the interesting point is that the probabilistic interpretations of
the N + 1-level chain gives rise to a very particular probabilistic structure:
the obtuse random variables.
7.3.1 Structure of Higher Level Chains
Let H be any separable Hilbert space. Let us fix a particular Hilbertian
basis (χi )i∈N ∪{0} for the Hilbert space H, where we assume (for notational
purposes) that 0 6∈ N . This particular choice of notations is motivated both
by physical interpretations (we see the χi , i ∈ N , as representing, for example,
different possible excited states of a quantum system, the vector χ0 represents
the “ground state” of the quantum system) and by a mathematical necessity
(for defining a countable tensor product of Hilbert space one needs to specify
one vector in each Hilbert space).
N
Definition 7.27. Let TΦ be the tensor product N∗ H with respect to the
stabilizing sequence χ0 . In other words, an orthonormal basis of TΦ is given
by the family {χσ ; σ ∈ P} where
– the set P = P(N∗ , N ) is the set of finite subsets {(n1 , i1 ), . . . , (nk , ik )} of
N∗ ×N such that the ni ’s are mutually different. Another way to describe the
set P is to identify it to the set of sequences (σn )n∈N∗ with values in N ∪ {0}
which take a value different from 0 only finitely many times;
– χσ denotes the vector
χ0 ⊗ . . . ⊗ χ0 ⊗ χi1 ⊗ χ0 ⊗ . . . ⊗ χ0 ⊗ χi2 ⊗ . . .
where χi1 appears in the n1 -th copy of H, where χi2 appears in the n2 -th
copy of H etc.
The physical signification of this basis is easy to understand: we have a
chain of quantum systems, indexed by N∗ . The space TΦ is the state space of
this chain, the vector χσ with σ = {(n1 , i1 ), . . . , (nk , ik )} represents the state
in which exactly k sites are excited: the site n1 in the state χi1 , the site n2
in the state χi2 etc, all the other sites being in their ground state χ0 .
Definition 7.28. This particular choice of a basis gives TΦ the same particular structure as the spin chain. If we denote by TΦn] the space generated by
the χσ such that σ ⊂ {1, . . . , n} × N and by TΦ[m the one generated by the
7 QUANTUM PROBABILITY
27
χσ such that σ ⊂ {m, m+1, . . .}×N , we get an obvious natural isomorphism
between TΦ and TΦn−1] ⊗ TΦ[n given by
[f ⊗ g](σ) = f (σ ∩ {1, . . . , n − 1} × N ) g (σ ∩ {n, . . .} × N ) .
Definition 7.29. Put {aij ; i, j ∈ N ∪ {0}} to be the natural basis of B(H),
that is,
aij (χk ) = δi,k χj
for all i, j, k ∈ N ∪ {0}. We denote by aij (n) the natural ampliation of the
operator aij to TΦ which acts on the copy number n as aij and which acts as
the identity on the other copies. That is, in terms of the basis χσ ,
aij (n)χσ = 1l(n,i)∈σ χσ\(n,i)∪(n,j)
if neither i nor j is zero, and
ai0 (n)χσ = 1l(n,i)∈σ χσ\(n,i) ,
a0j (n)χσ = 1ln6∈σ χσ∪(n,j) ,
a00 (n)χσ = 1ln6∈σ χσ ,
where n 6∈ σ actually means “for all i in N , (n, i) 6∈ N ”; we could equivalently
use the notation (n, 0) ∈ σ instead of n 6∈ σ, having in mind the interpretation
of σ as a sequence in N with finitely many non-zero terms.
In the case N = {1, . . . , N } we say that TΦ is Toy Fock space with multiplicity N + 1. In the case N = N∗ we say that TΦ is the Toy Fock space with
infinite multiplicity.
7.3.2 Obtuse Systems
The probabilistic interpretations of higher level chains is carried, in a natural
way, by particular random variables, the obtuse random variables. We shall
see in the next subsections that these random variables have very interesting
algebraic properties which characterize their behavior (and which encodes
the behavior of their continuous-time limit). Before entering into all those
properties, we start with obtuse systems.
Definition 7.30. Let N ∈ N∗ be fixed. An obtuse system in RN is a family
of N + 1 vectors v1 , . . . , vN +1 such that
hvi , vj i = −1
for all i 6= j. In that case we put
28
Stéphane ATTAL
vbi =
1
∈ RN +1 ,
vi
so that
hb
vi , vbj i = 0
for all i 6= j. They then form an orthogonal basis of RN +1 . We put
pi =
1
2
kb
vi k
=
1
1 + kvi k
2
,
for i = 1, . . . N + 1.
Lemma 7.31. With the notations above, we have
N
+1
X
pi = 1
(7.4)
p i vi = 0 .
(7.5)
i=1
and
N
+1
X
i=1
Proof. We have, for all j,
*N +1
+
X
1
2
pi vbi , vbj = pj kb
vj k = 1 =
, vbj .
0
i=1
As the vbj ’s form a basis, this means that
N
+1
X
pi vbi =
i=1
1
.
0
This gives the two announced equalities. t
u
Lemma 7.32. With the notations above, we also have
N
+1
X
pi |vi ihvi | = IRN .
(7.6)
i=1
√
Proof. As the vectors ( pi vbi )i∈{1,...,N +1} form an orthonormal basis of RN +1
we have
N
+1
X
IRN +1 =
pi |b
vi ihb
vi | .
i=1
Now, for all i = 1, . . . , N + 1, put
7 QUANTUM PROBABILITY
u=
29
1
0
and
vei =
0
,
vi
so that vbi = u + vei . We get
IRN +1 =
N
+1
X
pi |u + vei ihu + vei |
i=1
=
N
+1
X
pi |uihu| +
N
+1
X
i=1
pi |uihe
vi | +
N
+1
X
i=1
pi |e
vi ihu| +
N
+1
X
i=1
pi |e
vi ihe
vi | .
i=1
Using (7.4) and (7.5), this simplifies into
IRN +1 = |uihu| +
N
+1
X
pi |e
vi ihe
vi | .
i=1
In particular we have
N
+1
X
pi |vi ihvi | = IRN ,
i=1
that is, the announced equality. t
u
Let us consider two examples (that we shall meet again later in this section). On R2 , the 3 vectors
1
−1
−1
v1 =
,
v2 =
,
v3 =
0
1
−2
form an obtuse system of R2 . The associated pi ’s are then respectively
p1 =
1
,
2
p2 =
1
,
3
p3 =
1
.
6
We shall be interested also in another example. Let h > 0 be a parameter,
which shall be though of as small. On R2 , associated to the probabilities
1
,
2
p2 = h ,
v2 =
−1
1−2h 1/2
p1 =
p3 =
1
− h,
2
the 3 vectors
1
v1 =
,
0
form an obtuse system of R2 .
h
−1
!
v3 =
−2
h
1−2h
!
1/2
30
Stéphane ATTAL
We end this subsection with some useful properties of obtuse systems.
First, we prove an independence property for obtuse systems.
Proposition 7.33. Every strict sub-family of an obtuse family is linearly
free.
Proof. Let {v1 , . . . , vN +1 } be an obtuse family of RN . Let us show that
{v1 , . . . , vN } is free, which would be enough for our claim.
PN −1
If we had vN = i=1 λi vi then, taking the scalar product with vN we
P
2
N −1
get kvN k = i=1 −λi , whereas taking the scalar product with vN +1 gives
PN −1
−1 = i=1 −λi , whence a contradiction. t
u
Now we prove a kind of uniqueness result for obtuse systems.
Proposition 7.34. Let {v1 , . . . , vN +1 } be an obtuse system of RN having
{p1 , . . . , pN +1 } as associated probabilities. Then the following assertions are
equivalent.
i) The family {w1 , . . . , wN +1 } is an obtuse system on RN with same respective
probabilities {p1 , . . . , pN +1 }.
ii) There exists an orthogonal operator U on RN such that wi = U vi , for all
i = 1, . . . , N + 1.
Proof. One direction is obvious. If wi = U vi , for all i = 1, . . . , N + 1 and for
some orthogonal operator U, then the scalars products hvi , vj i and hwi , wj i
are equal, for each pair (i, j). This shows that {w1 , . . . , wN +1 } is an obtuse
system with the same probabilities.
In the converse direction, if v1 , . . . , vN +1 and w1 , . . . , wN +1 are obtuse
systems associated to the same probabilities p1 , . . . , pN +1 , then
hvi , vj i = hwi , wj i
for all i, j. The sub-families {v1 , . . . , vN } and {w1 , . . . , wN } are bases of RN ,
by Proposition 7.33, and their elements have same respective norms. The map
U : RN → RN , such that U vi = wi for all i = 1, . . . , N , is thus orthogonal,
by the conservation of scalar products. Finally, the vector vN +1 is a certain
linear combination of the vi ’s, i = 1, . . . , N , but wN +1 is the same linear
combination of the wi ’s, i = 1, . . . , N , by the conservation of scalar products.
Hence U vN +1 is equal to wN +1 and the proposition is proved. t
u
7.3.3 Obtuse Random Variables
Obtuse systems are strongly related to a certain class of random variables.
7 QUANTUM PROBABILITY
31
Definition 7.35. Consider a random variable X, with values in RN . We
shall denote by X 1 , . . . , X N the coordinates of X in RN . We say that X is
centered if its expectation is 0, that is, if E[X i ] = 0 for all i. We say that X
is normalized if its covariance matrix is I, that is, if
cov(X i , X j ) = E[X i X j ] − E[X i ] E[X j ] = δi,j ,
for all i, j = 1, . . . N .
Definition 7.36. Consider a random variable X, with values in RN , which
can take only N + 1 different non-null values v1 , . . . , vN +1 , with strictly
positive probability p1 , . . . , pN +1 , respectively. We consider the canonical
version of X, that is, we consider the probability space (Ω, F, P) where
Ω = {1, . . . , N + 1}, where F is the full σ-algebra of Ω, where the probability measure P is given by P ({i}) = pi and the random variable X is given
by X(i) = vi , for all i ∈ Ω. The coordinates of vi are denoted by vik , for
k = 1, . . . , N , so that X k (i) = vik .
In the same way as previously, we put
1
vbi =
∈ RN +1 ,
vi
for all i = 1, . . . , N + 1.
We shall also consider the deterministic random variable X 0 on (Ω, F, P),
e i be the random variable
which is always equal to 1. For i = 0, . . . , N let X
defined by
e i (j) = √pj X i (j)
X
for all i = 0, . . . , N and all j = 1, . . . , N + 1.
Proposition 7.37. With the notations above, the following assertions are
equivalent.
1) X is centered and normalized.
e i (j)
is an orthogonal matrix.
2) The (N + 1) × (N + 1)-matrix X
i,j
√
3) The (N + 1) × (N + 1)-matrix
pi vbij
is an orthogonal matrix.
i,j
4) The family {v1 , . . . , vN +1 } is an obtuse system with
pi =
1
,
1 + kvi k2
for all i = 1, . . . , N + 1.
Proof.
1) ⇒ 2): Since the random variable X is centered and normalized, each
component X i has a zero mean and the scalar product in L2 between two
components X i and X j is equal to δi,j . Hence, for all i in {1, . . . , N }, we get
32
Stéphane ATTAL
E[X i ] = 0
⇐⇒
N
+1
X
pk vki = 0 ,
(7.7)
k=1
and for all i, j = 1, . . . N ,
i
j
E[X X ] = δi,j
⇐⇒
N
+1
X
pk vki vkj = δi,j .
(7.8)
k=1
Now, using Eqs. (7.7) and (7.8), we get, for all i, j = 1, . . . , N
+1
E NX
D
e0 , X
e0 =
X
pk = 1 ,
k=1
+1
D
E NX
√ √
e0 , X
ei =
pk pk vki = 0 ,
X
k=1
+1
D
E NX
√
√
ei , X
ej =
X
pk vkj pk vki = δi,j .
k=1
The orthogonal character follows immediately.
e i (j)
2) ⇒ 1): Conversely, if the matrix X
is orthogonal, the scalar prodi,j
ucts of column vectors give the mean 0 and the covariance I for the random
variable X.
√
e i (j)
. Therepj vbij
is the transpose matrix of X
2) ⇔ 3): The matrix
i,j
i,j
fore, if one of these two matrices is orthogonal, its transpose matrix is orthogonal too.
√
3) ⇔ 4): The matrix
pj vbji i,j is orthogonal if and only if
√
pi vbi ,
√
pj vbj = δi,j ,
√
√
for all i, j = 1, . . . , N +1. On the other hand, the condition pi vbi , pi vbi =
√
√
pi vbi , pj vbj =
1 is equivalent to pi (1+kvi k2 ) = 1, whereas the condition
√ √
0 is equivalent to pi pj (1+hvi , vj i) = 0, that is, hvi , vj i = −1 . This gives
the result. t
u
Definition 7.38. Because of the equivalence between 1) and 4) above, the
random variables in RN which take only N + 1 different values with strictly
positive probability, which are centered and normalized, are called obtuse
random variables in RN .
7 QUANTUM PROBABILITY
33
7.3.4 Generic Character of Obtuse Random Variables
We shall here apply the properties of obtuse systems to obtuse random variable, in order to show that these random variables somehow generate all the
finitely supported probability distributions on RN .
First of all, an immediate consequence of Proposition 7.34 is that obtuse random variables on RN with a prescribed probability distribution
{p1 , . . . , pN +1 } are essentially unique.
Proposition 7.39. Let X be an obtuse random variable of RN with associated probabilities {p1 , . . . , pN +1 }. Then the following assertions are equivalent.
i) The random variable Y is an obtuse random variable on RN with same
probabilities {p1 , . . . , pN +1 }.
ii) There exists an orthogonal operator U on RN such that Y = U X in
distribution.
Having proved that uniqueness, we shall now prove that obtuse random
variables generate all the random variables (at least with finite support).
First of all, a rather simple remark which shows that the choice of taking
N +1 different values is the minimal one for centered and normalized random
variables in RN .
Proposition 7.40. Let X be a centered and normalized random variable in
Rd , taking n different values. Then we must have
n ≥ d + 1.
Proof. Let X be centered and normalized in Rd , taking the values v1 , . . . , vn
with probabilities p1 , . . . , pn and with n ≤ d, that is, n < d + 1. The relation
0 = E[X] =
n
X
pi vi
i=1
shows that Rank{v1 , . . . , vn } < n. On the other hand, the relation
ICd = E [|XihX|] =
n
X
pi |vi ihvi |
i=1
shows that Rank{v1 , . . . , vn } ≥ d. This gives the result. t
u
We can now state the theorem which shows how general, finitely supported,
random variables on Rd are generated by the obtuse ones. We concentrate
only on centered and normalized random variables, for they obviously generate all the others, up to an affine transform of Rd .
34
Stéphane ATTAL
Theorem 7.41. Let n ≥ d + 1 and let X be a centered and normalized random variable in Rd , taking n different values v1 , . . . , vn , with probabilities
p1 , . . . , pn respectively.
If Y is any obtuse random variable on Rn−1 associated to the probabilities p1 , . . . , pn , then there exists a partial isometry A from Rn−1 to Rd , with
Ran A = Rd , such that
X = AY
in distribution.
Proof. Assume that the obtuse random variable Y takes the values w1 , . . . , wn
in Rn−1 . The family {w1 , . . . , wn−1 } is linearly independent by Proposition
7.33, hence there exists a linear map A : Rn−1 → Rd such that A wi = vi for
all i < n. Now we have
!
X
X
X
pn vn = −
p i vi = −
pi A wi = A −
pi wi = pn A wn .
i<n
i<n
i<n
Hence the relation A wi = vi holds for all i ≤ n.
We have proved the relation X = A Y in distribution, with A being a
linear map from Rn−1 to Rd . The fact that X is normalized can be written
as E[XX ∗ ] = Id . But
E[XX ∗ ] = E[A Y Y ∗ A∗ ] = A E[Y Y ∗ ] A∗ = A In A∗ = A A∗ .
Hence A must satisfy A A∗ = Id , which is exactly saying that A is a partial
isometry with range Rd . t
u
7.3.5 Doubly-symmetric 3-tensors
Obtuse random variables are naturally associated to some 3-tensors with
particular symmetries. This is what we shall prove here.
Definition 7.42. A 3-tensor on Rn is an element of (RN )∗ ⊗ RN ⊗ RN , that
is, a linear map from RN to RN ⊗ RN . Coordinate-wise, it is represented by
N
a collection of coefficients (Sij
as
k )i,j,k=1,...,n . It acts on R
(S(x))ij =
n
X
k
Sij
kx .
k=1
We shall see below that obtuse random variables on RN have a naturally associated 3-tensor on RN +1 . Note that, because of our notation
choice X 0 , X 1 , . . . , X N , the 3-tensor is indexed by {0, 1, . . . , N } instead of
{1, . . . , N + 1}.
7 QUANTUM PROBABILITY
35
Proposition 7.43. Let X be an obtuse random variable in RN . Then there
exists a unique 3-tensor S on RN +1 such that
Xi Xj =
N
X
k
Sij
k X ,
(7.9)
k=0
for all i, j = 0, . . . , N . This 3-tensor S is given by
i
j
k
Sij
k = E[X X X ] ,
(7.10)
for all i, j, k = 0, . . . N .
Proof. As X is an obtuse random variable, that is, a centered and normalized random variable in RN taking exactly N +1 different values, the random
variables {X 0 , X 1 , . . . , X N } are orthonormal in L2 (Ω, F, P), hence they form
an orthonormal basis of L2 (Ω, F, P), for the latter space is N +1-dimensional.
These random variables being bounded, the products X i X j are still elements
of L2 (Ω, F, P), hence they can be written, in a unique way, as linear combinations of the X k ’s. As a consequence, there exists a unique 3-tensor S on
RN +1 such that
N
X
k
Sij
Xi Xj =
k X
k=0
for all i, j = 0, . . . , N . In particular we have
E[X i X j X k ] =
N
X
ij
Sij
l E[Xl Xk ] = Sk .
l=0
This shows Identity (7.10). t
u
This 3-tensor S has quite some symmetries, let us detail them.
Proposition 7.44. Let S be the 3-tensor associated to an obtuse random
variable X on RN . Then the 3-tensor S satisfies the following relations, for
all i, j, k, l = 0, . . . , N
Sij
(7.11)
0 = δij ,
N
X
Sij
k is symmetric in (i, j, k) ,
(7.12)
kl
Sij
m Sm is symmetric in (i, j, k, l) ,
(7.13)
m=0
Proof.
Relation (7.11) is immediate for
i
j
Sij
0 = E[X X ] = δij .
36
Stéphane ATTAL
Equation (7.12) comes directly from Formula (7.10) which shows a clear symmetry in (i, j, k).
By (7.9) we have
N
X
m
Xi Xj =
Sij
mX ,
m=0
whereas
Xk Xl =
N
X
n
Skl
n X .
n=0
Altogether this gives
N
X
kl
E Xi Xj Xk Xl =
Sij
m Sm .
m=0
But the left hand side is clearly symmetric in (i, j, k, l) and (7.13) follows.
t
u
7.3.6 The Main Diagonalization Theorem
We are going to leave for a moment the obtuse random variables and concentrate only on the symmetries we have obtained above. The relation (7.11)
is really specific to obtuse random variables, we shall leave it for a moment.
We concentrate on the relations (7.12) and (7.13) which have important consequences for the 3-tensor.
Definition 7.45. A 3-tensor S on RN +1 which satisfies (7.12) and (7.13) is
called a doubly-symmetric 3-tensor on RN +1 .
The main result concerning doubly-symmetric 3-tensors in RN +1 is that
they are the exact generalization for 3-tensors of normal matrices for 2tensors: they are exactly those 3-tensors which can be diagonalized in some
orthonormal basis of RN +1 .
Definition 7.46. A 3-tensor S on RN +1 is said to be diagonalizable in some
N
N
orthonormal basis (am )m=0 of RN +1 if there exist real numbers (λm )m=0
such that
N
X
S=
λm a∗m ⊗ am ⊗ am .
(7.14)
m=0
In other words
S(x) =
N
X
m=0
for all x ∈ RN +1 .
λm ham , xi am ⊗ am
(7.15)
7 QUANTUM PROBABILITY
37
Note that, as opposed to the case of 2-tensors (that is, matrices), the
“eigenvalues” λm are not completely determined by the representation (7.15).
Indeed, if we put e
am = sm am for all m, where the sm ’s are any modulus one
real numbers, then the e
am ’s still form an orthonormal basis of RN +1 and we
have
N
X
S(x) =
sm λm he
am , xi e
am ⊗ e
am .
m=0
Hence the λm ’s are only determined up to a sign, only their modulus is
determined by the representation (7.15).
Definition 7.47. Actually, there are more natural objects that can be associated to diagonalizable 3-tensors; they are the orthogonal families in RN .
Indeed, if S is diagonalizable as above, for all m such that λm 6= 0 put
vm = λm am . The family {vm ; m = 1, . . . , K} is then an orthogonal family
in RN +1 and we have
S(vm ) = vm ⊗ vm
for all m. In terms of the vm ’s, the decomposition (7.15) of S becomes
S(x) =
K
X
1
m=1
kvm k
2
hvm , xi vm ⊗ vm .
(7.16)
This is the form of diagonalization we shall retain for 3-tensors. Be aware
that in the above representation the vectors are orthogonal, but not normalized anymore. Also note that they only represent the eigenvectors of S
associated to non-vanishing eigenvalues.
We can now state the main theorem.
Theorem 7.48. A 3-tensor S on RN +1 is diagonalizable in some orthonormal basis if and only if it is doubly-symmetric.
More precisely, the formulas
V = v ∈ RN +1 \ {0} ; S(v) = v ⊗ v ,
and
S(x) =
X
v∈V
1
kvk
2
hv , xi v ⊗ v ,
establish a bijection between the set of orthogonal systems V in RN +1 \ {0}
and the set of complex doubly-symmetric 3-tensors S.
Proof. First step: let V = {vm ; m = 1, . . . , K} be an orthogonal family in
RN +1 \ {0}. Put
K
X
1
i
j
k
Sij
=
k
2 vm vm v m ,
kv
k
m
m=1
38
Stéphane ATTAL
for all i, j, k = 0, . . . , N . We shall check that S is a complex doubly-symmetric
3-tensor in RN . The symmetry of Sij
k in i, j, k is obvious from the definition.
This gives (7.12).
We have
N
X
kl
Sij
m Sm =
m=0
=
=
N
K
X
X
1
1
2
2
m=0 n,p=1 kvn k kvp k
K
X
1
1
2
2
n,p=1 kvn k kvp k
K
X
1
n=1 kvn k
2
vni vnj vnm vpm vpk vpl
vni vnj hvp , vn i vpk vpl
vni vnj vnk vnl
and the symmetry in i, j, k, l is obvious. This gives (7.13).
We have proved that the formula
S(x) =
X
v∈V
1
kvk
2
hv , xi v ⊗ v
(7.17)
defines a complex doubly-symmetric 3-tensor if V is any family of (nonvanishing) orthogonal vectors.
Second step: now given a complex doubly-symmetric 3-tensor S of the form
(7.17), we shall prove that the set V coincides with the set
b = {v ∈ CN \ {0} ; S(v) = v ⊗ v} .
V
Clearly, if y ∈ V we have by (7.17)
S(y) = y ⊗ y .
b Now, let v ∈ V.
b On one side we have
This proves that V ⊂ V.
S(v) = v ⊗ v ,
on the other side we have
S(v) =
X
z∈V
1
kzk
2
hz , vi z ⊗ z .
In particular, applying hy| ∈ V ∗ to both sides, we get
hy , vi v = hy , vi y
7 QUANTUM PROBABILITY
39
and thus either v is orthogonal to y or v = y. This proves that v is one of
the elements y of V, for it were orthogonal to all the y ∈ V we would get
v ⊗ v = S(v) = 0 and v would be the null vector.
We have proved that V coincides with the set
{v ∈ RN \ {0} ; S(v) = v ⊗ v} .
Third step: now we shall prove that all complex doubly-symmetric 3tensors S on RN +1 are diagonalizable in some orthonormal basis.
The property (7.12) indicates that the matrices
Sk = (Sij
k )i,j=0,...,N
are real symmetric hence they are all diagonalizable in some orthonormal
basis. The properties (7.12) and (7.13) imply
N
X
mk
Sim
=
j Sl
m=0
N
X
mk
Sim
l Sj .
m=0
In other words
(Sj Sl )ik = (Sl Sj )ik ,
for all i, k, all j, l. The matrices Sk commute pairwise. Thus the matrices
Sk can be simultaneously diagonalized: there exists an orthogonal matrix
U = (uij )i,j=0,··· ,N such that, for all k in {0, · · · , N },
Sk = U Dk U∗ ,
(7.18)
where the matrices Dk are diagonal: Dk = diag(λ0k , · · · , λN
k ). As a consequence, the coefficient Sij
can
be
written
as
k
Sij
k =
N
X
im jm
λm
u .
k u
m=0
Let us denote by am the mth column vector of U, that is, am = (ulm )l=0,··· ,N .
Moreover, we denote by λm the vector of λm
k , for k = 0, · · · , N . Since the
matrix U is orthogonal, the vectors am form an orthonormal basis of RN +1 .
We have
N
X
i
j
Sij
=
λm
k am am .
k
m=0
m
Our aim now is to prove that λ is proportional to am . To this end, we shall
use the symmetry properties of S. From the simultaneous reduction (7.18),
we get
Sj Sq = U Dj Dq U∗ .
40
Stéphane ATTAL
Therefore, we have
(Sj Sq )i,r =
N
X
mr
Sim
=
j Sq
m=0
N
X
m r
aim λm
j λq am .
m=0
In particular we have, for all p ∈ {0, . . . , N }
N
X
N
X
(Sj Sq )i,r aip λpj λpq arp =
ham , ap i hλm , λp i hλp , λm i hap , am i
m=0
i,j,q,r=0
4
= kλp k .
Note that
N
X
(7.19)
mr
Sim
j Sq
m=0
is also symmetric in (j, r). Applying this, the expression (7.19) is also equal
to
N
X
N
X
m j
i p p r
aim λm
r λq am ap λj λq ap =
i,j,q,r=0 m=0
=
N
X
hap , λm i hλm , λp i ham , λp i hap , am i
m=0
2
2
= |hap , λp i| kλp k .
This gives
|hap , λp i| = kλp k = kap k kλp k .
This is a case of equality in Cauchy-Schwarz inequality, hence there exists
µp ∈ R such that λp = µp ap , for all p = 0, . . . , N . This way, the 3-tensor S
can be written as
N
X
Sij
=
µm aim ajm akm .
(7.20)
k
m=0
In other words
S(x) =
N
X
µm ham , xi am ⊗ am .
m=0
We have obtained the orthonormal diagonalization of S. The proof is complete. t
u
7 QUANTUM PROBABILITY
41
7.3.7 Back to Obtuse Random Variables
The theorem above is a general diagonalization theorem for 3-tensors. For
the moment it does not take into account the relation (7.11). When we make
it enter into the game, we see obtuse systems appearing.
Theorem 7.49. Let S be a doubly-symmetric 3-tensor on RN +1 satisfying
also the relation
Sij
0 = δij
for all i, j = 0, . . . , N . Then the orthogonal system V such that
X
S(x) =
1
2
v∈V
kvk
hv , xi v ⊗ v
(7.21)
is made of exactly N + 1 vectors v1 , . . . , vN +1 , all of them satisfying vi0 = 1.
In particular the family of N + 1 vectors of RN , obtained by restricting the
vi ’s to their N last coordinates, forms an obtuse system in RN .
Proof. First assume that V = {v1 , . . . , vK }. By hypothesis, we have
Sij
k =
K
X
1
m=1 kvm k
2
i
j
k
vm
vm
vm
,
for all i, j, k = 0, . . . , N . With the supplementary property (7.11) we have in
particular
K
X
1
i
j
0
Sij
=
0
2 vm vm vm = δij
m=1 kvm k
for all i, j = 0, . . . , N .
Consider the orthonormal family of RN +1 made of the vectors em =
vm / kvm k. We have obtained above the relation
K
X
0
vm
|em ihem | = I ,
m=0
as matrices acting on RN +1 . The above is thus a spectral decomposition of
the identity matrix, this implies that the em ’s are exactly N + 1 vectors and
0
that all the vm
are equal to 1.
This proves the first part of the theorem. The last part concerning obtuse
systems is now obvious and was already noticed when we introduced obtuse
systems. t
u
In particular we have proved the following theorem.
42
Stéphane ATTAL
Theorem 7.50. The set of doubly-symmetric 3-tensors S on RN +1 which
satisfy also the relation
Sij
0 = δij
for all i, j = 0, . . . , N , is in bijection with the set of distributions of obtuse
random variables X on RN . The bijection is described by the following, with
the convention X 0 = 1l.
– The random variable X is the only (in distribution) random variable satisfying
N
X
k
X iX j =
Sij
kX ,
k=0
for all i, j = 1, . . . , N .
– The 3-tensor S is obtained by
i
j
k
Sij
k = E[X X X ] ,
for all i, j, k = 0, . . . , N .
In particular the different possible values taken by X in RN coincide with
the vectors wn ∈ RN , made of the last N coordinates of the eigenvectors vn
associated to S in the representation (7.21). The associated probabilities are
2
2
then pn = 1/(1 + kwn k ) = 1/kvn k .
7.3.8 Probabilistic Interpretations
Definition 7.51. Let X be an obtuse random variable in RN , with associated
3-tensor S and let (Ω, F, PS ) be the canonical space of X. Note that we have
added the dependency on S for the probability measure PS . The reason is that,
when changing the obtuse random variable X on RN , the canonical space Ω
and the canonical σ-field F do not change, only the canonical measure P does
change.
We have seen that the space L2 (Ω, F, PS ) is a N + 1-dimensional Hilbert
space and that the family {X 0 , X 1 , . . . , X N } is an orthonormal basis of that
space. Hence for every obtuse random variable X, with associated 3-tensor
S, we have a natural unitary operator
US : L2 (Ω, F, PS ) −→ CN +1
Xi
7−→ ei ,
where {e0 , . . . , eN } is the canonical orthonormal basis of CN +1 . The operator
US is called the canonical isomorphism associated to X.
Definition 7.52. On the space L2 (Ω, F, PS ), for each i = 0, . . . , N , we consider the multiplication operator
7 QUANTUM PROBABILITY
43
MX i : L2 (Ω, F, PS ) −→ L2 (Ω, F, PS )
Y
7−→
Xi Y ,
Definition 7.53. On the space CN +1 , with canonical basis {e0 , . . . , eN } we
consider the basic matrices aij , for i, j = 0, . . . , N defined by
aij ek = δi,k ej .
We shall see now that, when carried out on the same canonical space by
US , the obtuse random variables of RN admit a simple and compact matrix
representation in terms of their 3-tensor.
Theorem 7.54. Let X be an obtuse random variable on RN , with associated 3-tensor S and canonical isomorphism US . Then we have, for all
i, j = 0, . . . , N
N
X
Sijk ajk .
US MX i U∗S = a0i + ai0 +
(7.22)
j,k=1
for all i = 1, . . . , N .
Proof. We have, for any fixed i ∈ {1, . . . , N }, for all j = 0, . . . , N
US MX i U∗S ej = US MX i X j
= US X i X j
= US
N
X
k
Sij
kX
k=0
=
N
X
Sij
k ek .
k=0
Hence the operator US MX i U∗S has the same action on the orthonormal basis
{e0 , . . . , eN } as the operator
N
X
j
Sij
k ak .
j,k=0
Using the symmetries of S (Proposition 7.44) and the identity (7.11), we get
US M
Xi
U∗S
=
N
X
Si0
k
a0k
k=0
= ai0 + a0i +
+
N
X
Sij
0
aj0
+
j=1
N
X
j,k=1
This proves the representation (7.22). t
u
j
Sjk
i ak .
N
X
j,k=1
j
Sij
k ak
44
Stéphane ATTAL
Once again this is a remarkable point that we get here. All these different
obtuse random variables X of RN (and by the way, all the random variables in
RN with finite support) can be represented as very simple linear combinations
of the basic operators aij . As we have discussed above, all the probabilistic
properties of X (law, independence, functional calculus etc.) are carried by
these linear combinations of basic operators.
Let us check how this works with our examples. We first start with the
obtuse random variable X in R2 taking the values
1
−1
−1
v1 =
,
v2 =
,
v3 =
,
0
1
−2
with respective probabilities
p1 =
1
,
2
p2 =
1
,
3
p3 =
1
.
6
Adding a third coordinate 1 to each vector, we get the following orthogonal
system of R3 :
 
 
 
1
1
1
vb3 = −1 .
vb2 = −1 ,
vb1 = 1 ,
−2
1
0
We compute the matrices |b
vi ihb
vi | and get respectively


110
|b
v1 ihb
v1 | = 1 1 0
000


1 −1 1
|b
v2 ihb
v2 | = −1 1 −1
1 −1 1


1 −1 −2
|b
v3 ihb
v3 | = −1 1 2  .
−2 2 4
The matrices Sk , of multiplication by X k , are given by
X
i i i
Sij
p m vm
vj vk .
k =
m=02
This gives
7 QUANTUM PROBABILITY
45


100
S0 = 0 1 0
001


01 0
S1 = 1 0 0 
0 0 −1


0 0 1
S2 = 0 0 −1 .
1 −1 −1
These are the matrices associated to the multiplication operators by X 0 ,
X and X 2 , respectively. We recover facts that can be easily checked :
1
S0 = I,
(X 1 )2 = X 0 ,
X 1 X 2 = −X 2 ,
(X 2 )2 = X 0 − X 1 − X 2 .
As a consequence, the two operators
X1 = a10 + a01 − a22 ,
X2 = a20 + a02 − a12 − a21 − a22
have exactly the same probabilistic properties, in the sense of Quantum Probability on C3 in the state |e0 ihe0 |, as the pair of random variables (X 1 , X 2 ).
Let us now compute the 3-tensor associated to our second example, that
is the obtuse random X on R2 which takes the values
!
!
−1
−1
1
1/2
v3 =
v1 =
,
v2 =
h
1−2h 1/2
0
−2 1−2h
h
with respective probabilities
p1 =
1
,
2
p2 = h ,
p3 =
1
− h.
2
The same kind of computation as above gives


100
S0 = 0 1 0
001


01 0
S1 = 1 0 0 
0 0 −1


0 0
1

−1 
S2 = 0 0
.
1 −1 √ 1−4h
h(1−2h)
46
Stéphane ATTAL
At the level of this lecture we do not have the tools for computing rigorously the continuous-time limits of the random walks associated to the
two examples above, but we shall end this lecture by just describing these
continuous-time limits. In both cases consider the stochastic process
Ynh =
n √
X
h Xkh ,
k=0
where the Xkh are independent copies of the random variable X.
In the case of the first example, the process Y converges, when h tends to
0, to a two dimensional Brownian motion.
In the case of our second example, the limit process is a Brownian motion
in the first coordinate and a standard compensated poisson process in the
second coordinate.
Notes
Many parts of this chapter are inspired from the two reference books of
Parthasarathy [Par92] and Meyer [Mey93], and also from the course by Biane
[Bia95]. The two first references contain long developments concerning the
theory of quantum probability. Our approach in this chapter is closer to the
one of Parthasarathy.
The notion of Toy Fock space and its probabilistic interpretation as natural
space for all sequences of Bernoulli random variables, is due to P.-A. Meyer.
It first appeared in [Mey86]. The way we present it in this chapter is an
extended version, developed by Attal and Pautrat in [AP06].
Real obtuse random variables and their diagonalization theorem were introduced by Attal and Emery in [AÉ94].
References
[AÉ94] S. Attal and M. Émery. Équations de structure pour des martingales vectorielles. In Séminaire de Probabilités, XXVIII, volume 1583 of Lecture Notes
in Math., pages 256–278. Springer, Berlin, 1994.
[AP06] Stéphane Attal and Yan Pautrat. From repeated to continuous quantum
interactions. Ann. Henri Poincaré, 7(1):59–104, 2006.
[Bia95] Philippe Biane. Calcul stochastique non-commutatif. In Lectures on probability theory (Saint-Flour, 1993), volume 1608 of Lecture Notes in Math.,
pages 1–96. Springer, Berlin, 1995.
[Mey86] P.-A. Meyer. Éléments de probabilités quantiques. I–V. In Séminaire de
Probabilités, XX, 1984/85, volume 1204 of Lecture Notes in Math., pages
186–312. Springer, Berlin, 1986.
7 QUANTUM PROBABILITY
47
[Mey93] Paul-André Meyer. Quantum probability for probabilists, volume 1538 of
Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1993.
[Par92] K. R. Parthasarathy. An introduction to quantum stochastic calculus, volume 85 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1992.